15.3: Eukaryotic Transcription
Skills to Develop
- List the steps in eukaryotic transcription
- Discuss the role of RNA polymerases in transcription
- Compare and contrast the three RNA polymerases
- Explain the significance of transcription factors
Prokaryotes and eukaryotes perform fundamentally the same process of transcription, with a few key differences. The most important difference between prokaryotes and eukaryotes is the latter’s membrane-bound nucleus and organelles. With the genes bound in a nucleus, the eukaryotic cell must be able to transport its mRNA to the cytoplasm and must protect its mRNA from degrading before it is translated. Eukaryotes also employ three different polymerases that each transcribe a different subset of genes. Eukaryotic mRNAs are usually monogenic, meaning that they specify a single protein.
Initiation of Transcription in Eukaryotes
Unlike the prokaryotic polymerase that can bind to a DNA template on its own, eukaryotes require several other proteins, called transcription factors, to first bind to the promoter region and then help recruit the appropriate polymerase.
The Three Eukaryotic RNA Polymerases
The features of eukaryotic mRNA synthesis are markedly more complex those of prokaryotes. Instead of a single polymerase comprising five subunits, the eukaryotes have three polymerases that are each made up of 10 subunits or more. Each eukaryotic polymerase also requires a distinct set of transcription factors to bring it to the DNA template.
RNA polymerase I is located in the nucleolus, a specialized nuclear substructure in which ribosomal RNA (rRNA) is transcribed, processed, and assembled into ribosomes (Table \(\PageIndex{1}\)). The rRNA molecules are considered structural RNAs because they have a cellular role but are not translated into protein. The rRNAs are components of the ribosome and are essential to the process of translation. RNA polymerase I synthesizes all of the rRNAs except for the 5S rRNA molecule. The “S” designation applies to “Svedberg” units, a nonadditive value that characterizes the speed at which a particle sediments during centrifugation.
Table \(\PageIndex{1}\): Locations, Products, and Sensitivities of the Three Eukaryotic RNA Polymerases
| RNA Polymerase | Cellular Compartment | Product of Transcription | α-Amanitin Sensitivity |
|---|---|---|---|
| I | Nucleolus | All rRNAs except 5S rRNA | Insensitive |
| II | Nucleus | All protein-coding nuclear pre-mRNAs | Extremely sensitive |
| III | Nucleus | 5S rRNA, tRNAs, and small nuclear RNAs | Moderately sensitive |
RNA polymerase II is located in the nucleus and synthesizes all protein-coding nuclear pre-mRNAs. Eukaryotic pre-mRNAs undergo extensive processing after transcription but before translation. For clarity, this module’s discussion of transcription and translation in eukaryotes will use the term “mRNAs” to describe only the mature, processed molecules that are ready to be translated. RNA polymerase II is responsible for transcribing the overwhelming majority of eukaryotic genes.
RNA polymerase III is also located in the nucleus. This polymerase transcribes a variety of structural RNAs that includes the 5S pre-rRNA, transfer pre-RNAs (pre-tRNAs), and small nuclear pre- RNAs . The tRNAs have a critical role in translation; they serve as the adaptor molecules between the mRNA template and the growing polypeptide chain. Small nuclear RNAs have a variety of functions, including “splicing” pre-mRNAs and regulating transcription factors.
A scientist characterizing a new gene can determine which polymerase transcribes it by testing whether the gene is expressed in the presence of a particular mushroom poison, α-amanitin (table above). Interestingly, α-amanitin produced by Amanita phalloides , the Death Cap mushroom, affects the three polymerases very differently. RNA polymerase I is completely insensitive to α-amanitin, meaning that the polymerase can transcribe DNA in vitro in the presence of this poison. In contrast, RNA polymerase II is extremely sensitive to α-amanitin, and RNA polymerase III is moderately sensitive. Knowing the transcribing polymerase can clue a researcher into the general function of the gene being studied. Because RNA polymerase II transcribes the vast majority of genes, we will focus on this polymerase in our subsequent discussions about eukaryotic transcription factors and promoters.
Structure of an RNA Polymerase II Promoter
Eukaryotic promoters are much larger and more complex than prokaryotic promoters, but both have a TATA box. For example, in the mouse thymidine kinase gene, the TATA box is located at approximately -30 relative to the initiation (+1) site (Figure \(\PageIndex{1}\)). For this gene, the exact TATA box sequence is TATAAAA, as read in the 5' to 3' direction on the nontemplate strand. This sequence is not identical to the E. coli TATA box, but it conserves the A–T rich element. The thermostability of A–T bonds is low and this helps the DNA template to locally unwind in preparation for transcription.
 Would you expect the bacteria to transcribe the gene?
The mouse genome includes one gene and two pseudogenes for cytoplasmic thymidine kinase. Pseudogenes are genes that have lost their protein-coding ability or are no longer expressed by the cell. These pseudogenes are copied from mRNA and incorporated into the chromosome. For example, the mouse thymidine kinase promoter also has a conserved CAAT box (GGCCAATCT) at approximately -80. This sequence is essential and is involved in binding transcription factors. Further upstream of the TATA box, eukaryotic promoters may also contain one or more GC-rich boxes (GGCG) or octamer boxes (ATTTGCAT). These elements bind cellular factors that increase the efficiency of transcription initiation and are often identified in more “active” genes that are constantly being expressed by the cell.
Transcription Factors for RNA Polymerase II
The complexity of eukaryotic transcription does not end with the polymerases and promoters. An army of basal transcription factors, enhancers, and silencers also help to regulate the frequency with which pre-mRNA is synthesized from a gene. Enhancers and silencers affect the efficiency of transcription but are not necessary for transcription to proceed. Basal transcription factors are crucial in the formation of a preinitiation complex on the DNA template that subsequently recruits RNA polymerase II for transcription initiation.
The names of the basal transcription factors begin with “TFII” (this is the transcription factor for RNA polymerase II) and are specified with the letters A–J. The transcription factors systematically fall into place on the DNA template, with each one further stabilizing the preinitiation complex and contributing to the recruitment of RNA polymerase II.
The processes of bringing RNA polymerases I and III to the DNA template involve slightly less complex collections of transcription factors, but the general theme is the same. Eukaryotic transcription is a tightly regulated process that requires a variety of proteins to interact with each other and with the DNA strand. Although the process of transcription in eukaryotes involves a greater metabolic investment than in prokaryotes, it ensures that the cell transcribes precisely the pre-mRNAs that it needs for protein synthesis.
Evolution Connection: The Evolution of Promoters
The evolution of genes may be a familiar concept. Mutations can occur in genes during DNA replication, and the result may or may not be beneficial to the cell. By altering an enzyme, structural protein, or some other factor, the process of mutation can transform functions or physical features. However, eukaryotic promoters and other gene regulatory sequences may evolve as well. For instance, consider a gene that, over many generations, becomes more valuable to the cell. Maybe the gene encodes a structural protein that the cell needs to synthesize in abundance for a certain function. If this is the case, it would be beneficial to the cell for that gene’s promoter to recruit transcription factors more efficiently and increase gene expression.
Scientists examining the evolution of promoter sequences have reported varying results. In part, this is because it is difficult to infer exactly where a eukaryotic promoter begins and ends. Some promoters occur within genes; others are located very far upstream, or even downstream, of the genes they are regulating. However, when researchers limited their examination to human core promoter sequences that were defined experimentally as sequences that bind the preinitiation complex, they found that promoters evolve even faster than protein-coding genes.
It is still unclear how promoter evolution might correspond to the evolution of humans or other higher organisms. However, the evolution of a promoter to effectively make more or less of a given gene product is an intriguing alternative to the evolution of the genes themselves. 1
Promoter Structures for RNA Polymerases I and III
In eukaryotes, the conserved promoter elements differ for genes transcribed by RNA polymerases I, II, and III. RNA polymerase I transcribes genes that have two GC-rich promoter sequences in the -45 to +20 region. These sequences alone are sufficient for transcription initiation to occur, but promoters with additional sequences in the region from -180 to -105 upstream of the initiation site will further enhance initiation. Genes that are transcribed by RNA polymerase III have upstream promoters or promoters that occur within the genes themselves.
Eukaryotic Elongation and Termination
Following the formation of the preinitiation complex, the polymerase is released from the other transcription factors, and elongation is allowed to proceed as it does in prokaryotes with the polymerase synthesizing pre-mRNA in the 5' to 3' direction. As discussed previously, RNA polymerase II transcribes the major share of eukaryotic genes, so this section will focus on how this polymerase accomplishes elongation and termination.
Although the enzymatic process of elongation is essentially the same in eukaryotes and prokaryotes, the DNA template is more complex. When eukaryotic cells are not dividing, their genes exist as a diffuse mass of DNA and proteins called chromatin. The DNA is tightly packaged around charged histone proteins at repeated intervals. These DNA–histone complexes, collectively called nucleosomes, are regularly spaced and include 146 nucleotides of DNA wound around eight histones like thread around a spool.
For polynucleotide synthesis to occur, the transcription machinery needs to move histones out of the way every time it encounters a nucleosome. This is accomplished by a special protein complex called FACT , which stands for “facilitates chromatin transcription.” This complex pulls histones away from the DNA template as the polymerase moves along it. Once the pre-mRNA is synthesized, the FACT complex replaces the histones to recreate the nucleosomes.
The termination of transcription is different for the different polymerases. Unlike in prokaryotes, elongation by RNA polymerase II in eukaryotes takes place 1,000–2,000 nucleotides beyond the end of the gene being transcribed. This pre-mRNA tail is subsequently removed by cleavage during mRNA processing. On the other hand, RNA polymerases I and III require termination signals. Genes transcribed by RNA polymerase I contain a specific 18-nucleotide sequence that is recognized by a termination protein. The process of termination in RNA polymerase III involves an mRNA hairpin similar to rho-independent termination of transcription in prokaryotes.
Summary
Transcription in eukaryotes involves one of three types of polymerases, depending on the gene being transcribed. RNA polymerase II transcribes all of the protein-coding genes, whereas RNA polymerase I transcribes rRNA genes, and RNA polymerase III transcribes rRNA, tRNA, and small nuclear RNA genes. The initiation of transcription in eukaryotes involves the binding of several transcription factors to complex promoter sequences that are usually located upstream of the gene being copied. The mRNA is synthesized in the 5' to 3' direction, and the FACT complex moves and reassembles nucleosomes as the polymerase passes by. Whereas RNA polymerases I and III terminate transcription by protein- or RNA hairpin-dependent methods, RNA polymerase II transcribes for 1,000 or more nucleotides beyond the gene template and cleaves the excess during pre-mRNA processing.
 Would you expect the bacteria to transcribe the gene?
- Answer
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No. Prokaryotes use different promoters than eukaryotes.
Footnotes
- 1 H Liang et al., “Fast evolution of core promoters in primate genomes,” Molecular Biology and Evolution 25 (2008): 1239–44.
Glossary
- CAAT box
- (GGCCAATCT) essential eukaryotic promoter sequence involved in binding transcription factors
- FACT
- complex that “facilitates chromatin transcription” by disassembling nucleosomes ahead of a transcribing RNA polymerase II and reassembling them after the polymerase passes by
- GC-rich box
- (GGCG) nonessential eukaryotic promoter sequence that binds cellular factors to increase the efficiency of transcription; may be present several times in a promoter
- Octamer box
- (ATTTGCAT) nonessential eukaryotic promoter sequence that binds cellular factors to increase the efficiency of transcription; may be present several times in a promoter
- preinitiation complex
- cluster of transcription factors and other proteins that recruit RNA polymerase II for transcription of a DNA template
- small nuclear RNA
- molecules synthesized by RNA polymerase III that have a variety of functions, including splicing pre-mRNAs and regulating transcription factors


9.1: Composition Variables
A composition variable is an intensive property that indicates the relative amount of a particular species or substance in a phase.
9.1.1 Species and substances
We sometimes need to make a distinction between a species and a substance. A species is any entity of definite elemental composition and charge and can be described by a chemical formula, such as H\(_2\)O, H\(_3\)O\(^+\), NaCl, or Na\(^+\). A substance is a species that can be prepared in a pure state (e.g., N\(_2\) and NaCl). Since we cannot prepare a macroscopic amount of a single kind of ion by itself, a charged species such as H\(_3\)O\(^+\) or Na\(^+\) is not a substance. Chap. 10 will discuss the special features of mixtures containing charged species.
9.1.2 Mixtures in general
The mole fraction of species \(i\) is defined by \begin{gather} \s{ x_i \defn \frac{n_i}{\sum_j n_j} \qquad \tx{or} \qquad y_i \defn \frac{n_i}{\sum_j n_j} } \tag{9.1.1} \cond{(\(P{=}1\))} \end{gather} where \(n_i\) is the amount of species \(i\) and the sum is taken over all species in the mixture. The symbol \(x_i\) is used for a mixture in general, and \(y_i\) is used when the mixture is a gas.
The mass fraction , or weight fraction, of species \(i\) is defined by \begin{gather} \s{ w_i \defn \frac{m(i)}{m} = \frac{n_i M_i}{\sum_j n_j M_j} } \tag{9.1.2} \cond{(\(P{=}1\))} \end{gather} where \(m(i)\) is the mass of species \(i\) and \(m\) is the total mass.
The concentration , or molarity, of species \(i\) in a mixture is defined by \begin{gather} \s{ c_i \defn \frac{n_i}{V} } \tag{9.1.3} \cond{(\(P{=}1\))} \end{gather} The symbol M is often used to stand for units of mol L\(^{-1}\), or mol dm\(^{-3}\). Thus, a concentration of \(0.5\units{M}\) is \(0.5\) moles per liter, or \(0.5\) molar.
Concentration is sometimes called “amount concentration” or “molar concentration” to avoid confusion with number concentration (the number of particles per unit volume). An alternative notation for \(c\A\) is [A].
A binary mixture is a mixture of two substances.
9.1.3 Solutions
A solution , strictly speaking, is a mixture in which one substance, the solvent , is treated in a special way. Each of the other species comprising the mixture is then a solute . The solvent is denoted by A and the solute species by B, C, and so on. (Some chemists denote the solvent by subscript \(1\) and use \(2\), \(3\), and so on for solutes.) Although in principle a solution can be a gas mixture, in this section we will consider only liquid and solid solutions.
We can prepare a solution of varying composition by gradually mixing one or more solutes with the solvent so as to continuously increase the solute mole fractions. During this mixing process, the physical state (liquid or solid) of the solution remains the same as that of the pure solvent. When the sum of the solute mole fractions is small compared to \(x\A\) (i.e., \(x\A\) is close to unity), the solution is called dilute . As the solute mole fractions increase, we say the solution becomes more concentrated .
Mole fraction, mass fraction, and concentration can be used as composition variables for both solvent and solute, just as they are for mixtures in general. A fourth composition variable, molality, is often used for a solute. The molality of solute species B is defined by \begin{gather} \s{ m\B \defn \frac{n\B}{m(\tx{A})} } \tag{9.1.4} \cond{(solution)} \end{gather} where \(m(\tx{A})=n\A M\A\) is the mass of solvent. The symbol m is sometimes used to stand for units of mol kg\(^{-1}\), although this should be discouraged because m is also the symbol for meter. For example, a solute molality of \(0.6\units{m}\) is \(0.6\) moles of solute per kilogram of solvent, or \(0.6\) molal.
9.1.4 Binary solutions
We may write simplified equations for a binary solution of two substances, solvent A and solute B. Equations 9.1.1–9.1.4 become
\begin{gather} \s{ x\B = \frac{n\B}{n\A + n\B} } \tag{9.1.5} \cond{(binary solution)} \end{gather} \begin{gather} \s{ w\B = \frac{n\B M\B}{n\A M\A + n\B M\B} } \tag{9.1.6} \cond{(binary solution)} \end{gather} \begin{gather} \s{ c\B = \frac{n\B}{V} = \frac{n\B \rho}{n\A M\A + n\B M\B} } \tag{9.1.7} \cond{(binary solution)} \end{gather} \begin{gather} \s{ m\B = \frac{n\B}{n\A M\A} } \tag{9.1.8} \cond{(binary solution)} \end{gather} The right sides of Eqs. 9.1.5–9.1.8 express the solute composition variables in terms of the amounts and molar masses of the solvent and solute and the density \(\rho\) of the solution.
To be able to relate the values of these composition variables to one another, we solve each equation for \(n\B\) and divide by \(n\A\) to obtain an expression for the mole ratio \(n\B/n\A\): \begin{gather} \s{\tx{from Eq. 9.1.5}} \tag{9.1.9} \qquad \s{\frac{n\B}{n\A} = \frac{x\B}{1-x\B}} \cond{(binary solution)} \end{gather} \begin{gather} \s{\tx{from Eq. 9.1.6}} \tag{9.1.10} \qquad \s{\frac{n\B}{n\A} = \frac{M\A w\B}{M\B(1-w\B)}} \cond{(binary solution)} \end{gather} \begin{gather} \s{\tx{from Eq. 9.1.7}} \tag{9.1.11} \qquad \s{\frac{n\B}{n\A} = \frac{M\A c\B}{\rho - M\B c\B}} \cond{(binary solution)} \end{gather} \begin{gather} \s{\tx{from Eq. 9.1.8}} \tag{9.1.12} \qquad \s{\frac{n\B}{n\A} = M\A m\B} \cond{(binary solution)} \end{gather} These expressions for \(n\B/n\A\) allow us to find one composition variable as a function of another. For example, to find molality as a function of concentration, we equate the expressions for \(n\B/n\A\) on the right sides of Eqs. 9.1.11 and 9.1.12 and solve for \(m\B\) to obtain \begin{equation} m\B = \frac{c\B}{\rho - M\B c\B} \tag{9.1.13} \end{equation}
A binary solution becomes more dilute as any of the solute composition variables becomes smaller. In the limit of infinite dilution, the expressions for \(n\B/n\A\) become: \begin{gather} \s{\begin{split} \frac{n\B}{n\A} & = x\B \cr & = \frac{M\A}{M\B}w\B \cr & = \frac{M\A}{\rho\A^*}c\B = V\mA^* c\B \cr & = \s{ M\A m\B } \end{split}} \tag{9.1.14} \cond{(binary solution at} \nextcond{infinite dilution)} \end{gather}
where a superscript asterisk (\({}^*\)) denotes a pure phase. We see that, in the limit of infinite dilution, the composition variables \(x\B\), \(w\B\), \(c\B\), and \(m\B\) are proportional to one another. These expressions are also valid for solute B in a multi solute solution in which each solute is very dilute; that is, in the limit \(x\A\ra 1\).
The rule of thumb that the molarity and molality values of a dilute aqueous solution are approximately equal is explained by the relation \(M\A c\B/\rho\A^*=M\A m\B\) (from Eq. 9.1.14), or \(c\B/\rho\A^* = m\B\), and the fact that the density \(\rho\A^*\) of water is approximately \(1\units{kg L\(^{-1}\)}\). Hence, if the solvent is water and the solution is dilute, the numerical value of \(c\B\) expressed in mol L\(^{-1}\) is approximately equal to the numerical value of \(m\B\) expressed in mol kg\(^{-1}\).
9.1.5 The composition of a mixture
We can describe the composition of a phase with the amounts of each species, or with any of the composition variables defined earlier: mole fraction, mass fraction, concentration, or molality. If we use mole fractions or mass fractions to describe the composition, we need the values for all but one of the species, since the sum of all fractions is unity.
Other composition variables are sometimes used, such as volume fraction, mole ratio, and mole percent. To describe the composition of a gas mixture, partial pressures can be used (Sec. 9.3.1).
When the composition of a mixture is said to be fixed or constant during changes of temperature, pressure, or volume, this means there is no change in the relative amounts or masses of the various species. A mixture of fixed composition has fixed values of mole fractions, mass fractions, and molalities, but not necessarily of concentrations and partial pressures. Concentrations will change if the volume changes, and partial pressures in a gas mixture will change if the pressure changes.


3.6: Some kinematic identities
Learning Objectives
- List of various kinematics equations and identities
In addition to the relations
\[D(v) = \sqrt{\frac{1+v}{1-v}}\]
and
\[v_c = \frac{v_1 + v_2}{1 + v_1 v_2}\]
the following identities can be handy. If stranded on a desert island you should be able to rederive them from scratch. Don’t memorize them.
\[v = \frac{D^2 - 1}{D^2 + 1}\]
\[\gamma = \frac{D^{-1} + D}{2}\]
\[v\gamma = \frac{D - D^{-1}}{2}\]
\[D(v)D(-v) = 1\]
\[\eta = \ln D\]
\[v = \tanh \eta\]
\[\gamma = \cosh \eta\]
\[v\gamma = \sinh \eta\]
\[\tanh (x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y}\]
\[D_c = D_1 D_2\]
\[\eta _c = \eta _1 + \eta _2\]
\[v_C \gamma _c = (v_1 + v_2)\gamma _1 \gamma _2\]
The hyperbolic trig functions are defined as follows:
\[\sinh x = \frac{e^x - e^{-x}}{2}\]
\[\cosh x = \frac{e^x + e^{-x}}{2}\]
\[\tanh x = \frac{\sinh x}{\cosh x}\]
Their inverses are built in to some calculators and computer software, but they can also be calculated using the following relations:
\[\sinh^{-1}x = \ln \left ( x + \sqrt{x^2 + 1} \right )\]
\[\cosh^{-1}x = \ln \left ( x + \sqrt{x^2 - 1} \right )\]
\[\tanh^{-1}x = \frac{1}{2}\ln \left ( \frac{1 + x}{1 - x} \right )\]
Their derivatives are, respectively, \(\left ( x^2 + 1 \right )^{-1/2}\), \(\left ( x^2 - 1 \right )^{-1/2}\) and \(\left ( 1 - x^2 \right )^{-1}\).


12.8: Gothic Notre Dame (Started 1163 CE)
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The Notre-Dame de Paris Cathedral (7.29) is located on the Ile de la Cite in Paris, France, and is one of the best examples of Gothic architecture as well as one of the most well-known churches in the world. The cathedral began in 1163 CE after building a unique roadway to transport building supplies, causing the demolition of several houses on the new site.
Gothic cathedrals have large walls of windows displaying colors from streaming sunlight, are encircled by a spider network of flying buttresses, and topped with ornamental gargoyles. Gothic architecture became a familiar feature of many great churches and castles in this period and led to towns competing to build the most significant and grandest cathedrals. Building a cathedral consumed the labor of surrounding towns, and occasionally took over 100 years to complete. Gothic cathedrals began to rise from the ground, stone by stone, to higher and higher heights, throughout the Western European Catholic region.
The architecture included unique structures, pointed arches, ribbed vaults, and flying buttresses. The pointed arches (7.30) allowed the height of the church to soar and large spaces for windows in these gothic churches instead of the restrictions found in the thick-walled Romanesque church. Because of the invention and architecture of ribbed vaulting (7.31), the architectural capabilities for taller churches and larger windows were possible. Using the strength of stone close to the roof was conceivable, with the construction of multiple barrel vaults intersecting in the middle.
As Gothic churches ascended in height and they constructed large openings for stained glass, however, with the additional height of the walls, the lateral forces of the wall began to push outward, causing the heavy stone load above to tumble. The design of the flying buttresses (7.32) was to redirect the forces from the pointed arches to the ground. They projected from the walls to a foundation, supporting the weight of the high walls by carrying the pressure of the lateral thrust and stabilizing the walls. The flying buttresses became an architectural part of the structure (7.33).
The cathedral was a tribute to Gothic architecture, and the layout of the church followed the basilica style. A nave is the central aisle in the building and usually ends in the rounded end to form the apse with additional aisles along each side of the nave separated by colonnades. The transept cuts across the aisle between the nave and apse, which gives the building its cross-like shape, the choir area is located where the nave and the transept intersect.
Basilica style refers to a very large open communal gathering place.
The cathedral was built in parts; the choir area from 1163 to 1177 CE, the high altar finished in 1218 CE, the transepts and nave by 1208 CE, and the western façade and finishing details all finally completed by 1240 CE. Over the years, many different architects worked on the design of the building resulting in distinctive styles and heights of different sections and towers. Each architect specialized in distinct aspects of the building; for example, the work of one architect was to oversee the assembly of the rose windows. Other architects focused on the great halls beneath the towers.
One of the many design modifications occurred while building the transepts, especially the gabled portal on the north transept, a change to replace the ordinary with a spectacular rose (7.34) window. The idea so successful they added a similar window to the southern transept. The two transept portals were decorated with images from different religious stories, images used to teach devotees during weekly religious gatherings.
The cathedral was one of the first buildings to use the flying buttresses with their arched exterior supports designed for the stability of the structure. Initially, the original architectural drawings did not include the flying buttresses; however, after construction began, the thin walls of the Gothic style building began to fracture. At the joining of the pointed arch, the forces of gravity started to push outward, which can lead to the walls collapsing. The architects designed the buttress supports around the building located at exact points of failure; the fracturing stopped. Other tall cathedrals were experiencing fracturing, triggering flying buttresses to become the standard in building during the Gothic era.
Notre Dame is beyond a church as it is also draped in art. Small statues (7.35) of religious figures and stories used as decoration everywhere, another method of teaching the illiterate population about the tenets of their religion. Initially, the ornate statues were decorated with bright paint and gold gilding. Other figures, the most famous are the chimeras (7.36) and gargoyles designed as waterspouts, were symbols of evil, meant to threaten people, and encourage proper religious adherence. Gargoyles (7.37) were first designed in the 13th century to disperse water from the roof away from the foundations of the building and used extensively across the roofline; they also became the guardians of the building.


7.2: Kepler's Laws
More than 20 years before Newton was born, a fellow named Johannes Kepler took a shot at explaining the orbits of the planets. He too posited that physical laws might be able to explain the motions, but didn’t possess the tools (mathematical and physical) at Newton’s disposal decades later (though admittedly, Newton did develop these tools for himself). Instead, what Kepler had were the remarkably detailed and accurate measurements of planetary motions made by an astronomer named Tycho Brahe, which he used to look for patterns in the motions. Amazingly, he found that the planets indeed moved with mathematical precision, and published his three laws of celestial motion, all of which are in exact accordance with the law of universal gravitation. While reading about his three laws, consider what a monumental accomplishment this was. Tycho Brahe's data detailed the motions of the planets as he viewed them from Earth (which itself is orbiting the sun).
Kepler’s First Law
Kepler's First Law: The paths of bodies trapped in orbits form closed ellipses, with the gravitating body at one of the foci.
The many elements of an ellipse and how an orbit fits into the picture are expressed in Figure 7.2.1 .
 A common way is to write the distance between the two bodies as a function of the angle that the line between them makes with the major (longer) axis:
Figure 7.2.2 – Polar Coordinate Description of Ellipse
The formula for the ellipse in these coordinates is:
\[ r\left(\theta\right)=\dfrac{a\left(1-e^2\right)}{1-e\cos\theta} \]
[ Note: It is also possible to measure the angle in the opposite direction (with \(\theta=0\) corresponding to the perihelion), in which case the denominator is a sum rather than a difference. ]
Circles are special cases of ellipses (eccentricity equal to zero), so naturally circular orbits are possible. Notice that if we plug in \(e=0\) above, we get the simple orbit equation \(r=a\). With a great deal of mathematics (first surmounted by Isaac Newton), one can can show that in fact this is a natural consequence of the inverse-square force law we already stated for gravitation. While we won't go quite so far as to perform this derivation, below we will make a closer examination of features of the ellipse (which we have expressed as a purely mathematical object here) in terms of physical quantities.
Kepler’s Second Law
Kepler's Second Law: An orbiting planet sweeps out equal areas in equal times during its orbit.
Kepler noticed that the planet moved faster when it was near the perihelion than when it was near the aphelion, and through painstaking examination of the data determined that in fact the amount of area the orbit sweeps out in a given period of time is the same everywhere in the orbit.
Figure 7.2.3 – Equal Areas in Equal Times
Note that while the diagram compares areas swept out through the perihelion and aphelion, the result applies to any part of the orbit – if we wait the same period of time, the area swept out will be the same: \(A_1=A_2\). We will soon look into the physical aspects of gravitational orbits that lead to this result, but again one can't help but marvel at what it must have taken to derive this remarkable discovery from the raw data.
Kepler’s Third Law
Kepler's Third Law: For every object orbiting the same gravitational source, the ratio of the cube of the semi-major axis of the orbital ellipse and the square of the orbital period is the same constant: \(\dfrac{a^3}{T^2} = constant\).
While the first law makes a general statement about all gravitational orbits, and the second law relates two different parts of a specific gravitational orbit, the third law gives a way of comparing different orbits of the same gravitating object (in Kepler's case, this gravitating object was the sun). Of the three laws, this one has the greatest practical value, because it means that even without knowing anything about the law of gravitation, one can make a prediction about one orbiting body based on observations of another orbiting body, if both are going around the same gravitating object. We will see what this mysterious constant is in terms of physical details, but to solve certain problems, knowing the actual value of the constant isn't necessary.
Example \(\PageIndex{1}\)
An astronomer notices that an asteroid is positioned such that the Earth is directly between it and the Sun. It has a roughly circular orbit (like the Earth), which is in the same plane and in the same direction as the Earth. This asteroid is far away from the Earth – about 8 times farther from the Earth than the distance separating the Earth and the Sun. These are all approximations, but about how long will this astronomer have to wait see these three bodies reach these same positions?
- Solution
-
For a circular orbit, the semi-major axis of the orbit is simply the radius of the orbit, and since the asteroid and Earth are both orbiting the same gravitational source (the sun), then Kepler's third law results in the same constant for both:
\[ \dfrac{R_{earth}^3}{T_{earth}^2} = constant = \dfrac{R_{asteroid}^3}{T_{asteroid}^2} \;\;\; \Rightarrow \;\;\; T_{asteroid}=\left(\dfrac{R_{asteroid}}{R_{earth}}\right)^{\frac{3}{2}}T_{earth} \nonumber \]
The period of the Earth's orbit is exactly 1 year, and the asteroid is 9 times farther from the sun as the Earth, so the time it will take the asteroid to come back to the same spot will be:
\[ T_{asteroid} = 27years \nonumber \]
Of course, in 27 years, the Earth will also be in the spot where it started, so this is our answer.
Note that if the question asked for the time that elapses before the sun, Earth, and asteroid are all aligned again (which is different from reaching their original positions), the answer is different: When they realign for the first time, the Earth will have completed one orbit plus a bit more, while the asteroid will have completed a fraction of an orbit. Let's call the angle that the asteroid moves through in that first year \(\Delta \theta\). The Earth catches up to it after a full revolution, so the angle the Earth moves through is:
\[ \theta_{earth} = 2\pi + \Delta \theta \nonumber \]
The Earth is moving 27 times as fast as the asteroid, so in this equal time frame the Earth has moved through an angle 27 times as great as the asteroid, which gives:
\[ 2\pi + \Delta \theta = 27\Delta \theta \;\;\; \Rightarrow \;\;\; \Delta \theta = \dfrac{2 \pi}{26} \nonumber \]
That is, the asteroid completes 1/26th of its orbit at the point when the Earth catches up to it. The asteroid's orbit takes 27 years, so the first alignment occurs at:
\[\Delta T = \dfrac{27years}{26} = 1 year, 14 days \nonumber \]
A nice application of Kepler’s 3rd law involves man-made satellites that orbit the Earth. Telecommunications satellites we like to remain at a single position in the sky, so that we don’t have to turn our satellite dishes to find them – we just point them in the right direction and leave it. To accomplish this, we need two things: The satellite has to be directly above the equator, and it has to be orbiting the Earth in the same direction that the Earth is rotating, with an orbital period of exactly 1 day. This is known as a geostationary orbit. We can use Kepler’s 3rd law to determine how high off the Earth’s surface this satellite needs to be.
Reconciling Kepler’s Laws with Universal Gravitation
There are a couple of things we can say about the physics of gravitational orbits. First, gravity is a conservative force, which means that the mechanical energy of the system is conserved. We don't yet know how to describe the potential energy due to the gravitational force (hopefully it is clear that our old "\(U\left(y\right) = mgy + U_o\)" treatment is no longer adequate, since this results in a constant force), but we will look at this in Section 7.3 . The point is that the mechanical energy is a "constant of the motion," which we can use, for example, to describe the speed of the orbiting body as a function of the distance from the gravitational source.
Consider the Earth + sun system. The fraction of this system's mass that belongs to the Earth is about \(3 \times 10^{-6}\), which means that the distance from the center of the sun to the center of mass of the system is this fraction multiplied by the (on average) 92 million miles separating the two bodies. The distance from the center of the sun to the center of mass of the system is therefore:
\[ r_{cm} = \left(3 \times 10^{-6}\right)\left(92 \times 10^6\;miles\right) = 276 \; miles\]
The radius of the sun is about 432,000 miles, so the center of mass of the system lies less than one tenth of one percent of the sun's radius from the sun's center. It's therefore a pretty good approximation to treat the center of the sun as a fixed point. [ Note: Even the center of mass of the Jupiter + sun system barely lies outside the sun's radius, even though Jupiter is much more massive than Earth, and is much farther away. ] The gravitational force is directed at this fixed point, so it constitutes a central force. As we found in Section 6.2 , central forces have the property of conserving the angular momentum of the system (since they produce no torque). We therefore conclude that like the mechanical energy, the angular momentum of an orbiting body is also a constant of the motion, when the gravitating body is significantly more massive than the orbiting body. [ Note: The angular momentum of the whole system is always conserved, even when the two masses are comparable, but in that case we can't treat one object as orbiting another stationary one, which means we have to consider the motion of both objects, complicating our picture. ]
Kepler's First Law
Showing that elliptical orbits are a direct result of the law of universal gravitation is a mathematical exercise that is somewhat beyond the scope of this work. While this derivation could nevertheless be included here, the value of doing so is minimal, and will therefore be left for the reader to explore in an upper-division treatment of classical mechanics. Instead, we will look at only small – but very instructive – aspects of Kepler's first law.
Never mind that we have no reason to expect that orbits will be elliptical... Why would we even expect them to be closed ? That is, it certainly isn't clear that when the polar angle \(\theta\) changes by \(2\pi\), that the orbiting body's distance from the gravitating body will be the same. It turns out, however, that this element of gravitational orbits is not hard to demonstrate. Start by applying Newton's second law to the orbiting body on which a net force due to gravity is acting:
\[ m\dfrac{d\overrightarrow v}{dt} = -\dfrac{GMm}{r^2} \widehat r \]
We can rewrite this in terms of how the velocity changes with the angle \(\theta\) by using the chain rule:
\[ m\dfrac{d\overrightarrow v}{d\theta}\dfrac{d\theta}{dt} = -\dfrac{GMm}{r^2} \widehat r \]
The angular momentum of the orbiting body can be written in terms of its mass, its angular velocity \(\omega = \dfrac{d\theta}{dt}\), and its distance from the fixed point:
\[ L = mr^2\omega \]
This angular momentum is a constant of the motion (i.e. it is conserved throughout the orbit), so we find that the rate at which the velocity vector changes with respect to \(\theta\) is a vector with a constant magnitude:
\[ \dfrac{d\overrightarrow v}{d\theta} = -\dfrac{GMm}{L} \widehat r \]
We can write the position unit vector in terms of the angle relative to a cartesian coordinate system, as we did in Equation 1.6.11 :
\[ \dfrac{d\overrightarrow v}{d\theta} = -\dfrac{GMm}{L} \left( \cos \theta \; \widehat i \ + \sin \theta \; \widehat j \right) \]
Integrating over the angle gives the velocity vector as a function of \(\theta\) (and an undetermined constant of integration \(\overrightarrow v_o\)):
From this result we can conclude that the magnitude and direction of the velocity return to the same value every time \(\theta\) changes by \(2\pi\). This means that the kinetic energy returns to its same value periodically as well. But the mechanical energy of this system is conserved, so the potential energy also returns to its value with the same periodicity. But (as we will see in the next section), the potential energy is defined by the separation of the two masses, so this separation also returns every time \(\theta\) changes by \(2\pi\). Well, if every time the angle changes by \(2\pi\) the orbiting body is the same distance away from the gravitating body, is moving at the same speed, and is moving in the same direction, then clearly its motion is being repeated – the orbit is closed.
Notice that if dependence on \(r\) in the law of gravitation was anything other than inverse-square, then the \(r\)'s would not cancel as they did in reaching Equation 7.2.6 (the angular momentum would still be the same constant of the motion, as the force would still be central), which would give an equation for the velocity that depends on both \(r\) and \(\theta\). This ruins the argument above, and the orbit would not be closed.
Digression: Orbits Are Not Quite Closed After All
As amazing as Newton's accomplishment was with his theory of gravity, roughly 230 years later, a fellow named Albert Einstein provided an improved theory, called the General Theory of Relativity. It had been known for some time that observations of the orbit of Mercury indicated that its orbit was in fact not closed. For a long time it was thought that the discrepancy was the result of an unseen planet or other gravitating body that was pulling Mercury off course, but Einstein's theory showed that the inverse-square theory of Newton, while a very good approximation, is not quite right, and his new theory predicted Mercury's motion perfectly.
While we are skipping the mathematics detailing how we get to the equation of the ellipse, we can still extract some information from Equation 7.2.8 that will be useful to us later. To simplify the discussion that follows, we will assume that we already know that the elliptical orbit is the result.
We haven't yet defined the orientation of the \(\left(x,y\right)\) coordinate system used in Equation 7.2.8 – so far we have only required that the origin be at the gravitating body. Let's choose the \(x\)-axis to lie along the major axis such that the point of maximum separation (aphelion) lies on the positive side of the \(x\)-axis (giving us Figure 7.2.2 ). With these axes, it is clear that at the point \(\theta = 0\), the velocity of the orbiting body is in the \(+\widehat j\) direction. Looking at Equation 7.2.8 , we see that this means that the constant vector \(\overrightarrow v_o\) must point parallel to the minor axis, otherwise it would give the body a component of velocity along the \(x\) direction. But which way does this constant vector point, \(+\widehat j\) or \(-\widehat j\)?
Consider the velocity of the orbiting body at the perihelion (\(\theta = \pi\)). In this case, the object is now moving in the \(-\widehat j\) direction, but because it is closer to the reference point at the gravitating body, it must be moving faster to conserve angular momentum. For the constant vector \(\overrightarrow v_o\) to make the orbiting body faster at (\(\theta = \pi\)) than at (\(\theta = 0\)), we must have:
We can relate the maximum and minimum speeds of the orbit using angular momentum conservation. Looking at Figure 7.2.1 , we see that the value of \(r_\bot\) for the aphelion is: \(a+ea = a\left(1+e\right)\). At the perihelion: \(r_\bot = a\left(1-e\right)\). Setting equal the angular momenta at these two positions in the orbit gives:
\[ L_{aphelion} = L_{perihelion} \;\;\; \Rightarrow \;\;\; mv_{min}\left[a\left(1+e\right)\right] = mv_{max}\left[a\left(1-e\right)\right] \;\;\; \Rightarrow \;\;\; v_{min} = v_{max}\left(\dfrac{1-e}{1+e}\right) \]
Using this result and Equations 7.2.9 , we can eliminate the troublesome \(v_o\), and get:
It will also be useful to have an expression for the angular momentum in terms of the masses, eccentricity, and major axis. To get this, multiply the first of the Equations 7.2.11 by the orbiting body's mass and \(r_\bot\) [for \(v_{max}\) this is: \(a\left(1-e\right)\)] to get:
Putting this back into Equation 7.2.1 simplifies it a bit in terms of physical constants, putting in a form that will be useful later:
\[ r\left(\theta\right)=\left(\dfrac{L^2}{GMm^2}\right)\dfrac{1}{1-e\cos\theta} \]
One last thing to note before moving on to Kepler's second law. Looking at Equation 7.2.8 , we see that if \(v_o\) happens to equal zero, then the velocity vector has a constant magnitude, and its direction is always perpendicular to \(\widehat r\) (which can be confirmed quickly by performing a dot product). So a non-zero constant of integration is responsible for making an otherwise circular orbit eccentric.
Example \(\PageIndex{2}\)
Show that the distance of closest approach (the perihelion distance) is given by:
\[ r_{min} = \dfrac{L^2}{GMm^2}\left(\dfrac{1}{1+e}\right) \nonumber \]
Do this in two different ways:
- Using calculus and Equation 7.2.13 . [Note: There are two angles that result in extrema, but only one gives a minimum.]
- Using the equation for angular momentum at \(r_{min}\) and one of the Equations 7.2.11 .
- Solution
-
a. We seek the minimum value of \(r\left(\theta\right)\), so we start by finding the value of \(\theta\) where this minimum occurs:
\[ 0 = \left(\dfrac{L^2}{GMm^2}\right)\dfrac{d}{d\theta}\left(\dfrac{1}{1-e\cos\theta}\right) \;\;\; \Rightarrow \;\;\; 0 = \dfrac{\sin\theta}{1-e\cos\theta} \;\;\; \Rightarrow \;\;\; \theta = 0 \;\; or\;\; \pi \nonumber \]
Note that \(\theta = 0\) minimizes the denominator, so it gives a maximum for \(r\), not a minimum. On the other hand, \(\theta = \pi\) maximizes the denominator, and therefore gives a minimum. This result makes sense when we look at how \(theta\) is defined in Figure 7.2.2 . Plugging \(\theta = \pi\) into Equation 7.2.13 gives the desired answer.
b. The angular momentum at the point closest approach will involve the maximum speed, since it is conserved throughout the orbit, so using the expression for the maximum speed in Equations 7.2.11 we get:
\[ L = mv_{max} r_{min} \;\;\; \Rightarrow \;\;\; r_{min} = \dfrac{L}{mv_{max}} = \dfrac{L}{m\left( \left(1+e\right)\dfrac{GMm}{L}\right)} = \dfrac{L^2}{GMm^2}\left(\dfrac{1}{1+e}\right) \nonumber\]
Kepler's Second Law
We look next at Kepler's equal-areas-swept-out-in-equal-times law. The geometry of measuring areas swept out of ellipses is impossibly difficult to do mathematically, but an infinitesimal amount of area swept out in an infinitesimal time period is something we can do. Figure 7.2.4 shows how we can mathematically describe the area swept out in an infinitesimal period of time. The amount swept out can be broken into two triangles. Okay, so the sides of the triangles are curved, but when the curves are infinitesimal in length, the amount they differ from straight lines is insignificant. The area of the two triangles are color-coded in the diagram. We notice that while both have infinitesimal areas, the orange triangle includes a product of two infinitesimal quantities. When the area is then divided by a small time span and the limit is taken as the time span goes to zero, the ratio of \(d\theta\) and \(dt\) approaches a finite value (specifically, the angular velocity \(\omega\) at that moment in time), but in the second term there is still another infinitesimal \(dr\) that goes to zero in the limit. In other words, the orange triangle contributes nothing to the area swept out in the infinitesimal time span \(dt\). With a little mathematical manipulation, we see that the rate at which area is swept out is the angular momentum of the orbiting body divided by twice its mass. We know that both the angular momentum and the mass remain constant for the orbit, so the rate at which area is swept out also remains constant. Kepler's second law is equivalent to angular momentum conservation.
Figure 7.2.4 – Kepler's Second Law Expresses Angular Momentum Conservation
Kepler's Third Law
It is quite straightforward to show that Kepler's third law holds for circular orbits, so let's do that first. The speed is constant for the entire orbit, it equals the circumference of the orbit divided by the time it takes for a full orbit:
We can use the fact that the gravitational force is causing centripetal acceleration to get the following expression for the square of the constant speed of the orbiting body:
\[ \dfrac{GMm}{R^2}=m\dfrac{v^2}{R} \;\;\; \Rightarrow \;\;\; v^2 = \dfrac{GM}{R} \]
Plugging Equation 7.2.14 into Equation 7.2.15 and doing some algebra gives Kepler's third law, with the semi-major axis equaling the radius of the circular orbit (zero eccentricity):
This gives us not only that the ratio is a constant, but specifically what the constant is. As we can now confirm, this constant depends only upon the mass of the gravitating body.
It's quite remarkable that this law holds equally well for elliptical orbits, where \(R\) is replaced by \(a\). We can show this by starting with a result we found from Kepler's second law. The rate at which area is swept out is constant, so the total area of the ellipse is this rate multiplied by the time of a full orbit. Reviewing our conic sections, we plug in the area of an ellipse, and get:
\[ area\;of\;ellipse = \pi \; ab = \dfrac{dA}{dt} T = \dfrac{L}{2m} T \]
The quantity \(b\) is the length of the semi-minor axis, which is related to the length of the semi-major axis in terms of the eccentricity:
\[ b^2 = a^2\left(1-e^2\right) \]
Squaring Equation 7.2.17 and eliminating \(b^2\) using Equation 7.2.18 gives us:
\[ \pi^2 a^4 \left(1-e^2\right) = \dfrac{L^2T^2}{4m^2}\]
We had the good foresight to derive Equation 7.2.12 , so plugging \(L^2\) from that equation into Equation 7.2.19 gives us our answer:
\[ \pi^2 a^4 \left(1-e^2\right) = \left[a \left(1-e^2\right) GMm^2 \right] \dfrac{T^2}{4m^2} \;\;\; \Rightarrow \;\;\; \dfrac{a^3}{T^2}= \dfrac{GM}{4\pi^2} \]
Example \(\PageIndex{3}\)
Attractive central forces that are not inverse-square do not produce closed orbits except when the orbit happens to be circular. Whenever there is a closed orbit, a result like Kepler's third law (which relates the orbit period to the radius of the circular orbit) will be the result. Derive the effective Kepler third law for an attractive central force that varies as the inverse-cube of the separation.
- Solution
-
Following the process used above for a gravitational circular orbit, we get:
\[ \left. \begin{array}{l} \dfrac{k}{R^3}=m\dfrac{v^2}{R} \\ v = \dfrac{2\pi R}{T} \end{array} \right\} \;\;\; \Rightarrow \;\;\; \boxed{\dfrac{R^4}{T^2} = \dfrac{k}{4\pi^2m} = constant} \nonumber \]


9.1: Introduction
The apparatus used by Faraday to demonstrate that magnetic fields can create currents is illustrated in Figure \(\PageIndex{1}\). When the switch is closed, a magnetic field is produced in the coil on the top part of the iron ring and transmitted to the coil on the bottom part of the ring. The galvanometer is used to detect any current induced in the coil on the bottom. It was found that each time the switch is closed, the galvanometer detects a current in one direction in the coil on the bottom. (You can also observe this in a physics lab.) Each time the switch is opened, the galvanometer detects a current in the opposite direction. Interestingly, if the switch remains closed or open for any length of time, there is no current through the galvanometer. Closing and opening the switch induces the current. It is the change in magnetic field that creates the current. More basic than the current that flows is the source voltage that causes it. The current is a result of an source voltage induced by a changing magnetic field , whether or not there is a path for current to flow.
An experiment easily performed and often done in physics labs is illustrated in Figure \(\PageIndex{2}\). A source voltage is induced in the coil when a bar magnet is pushed in and out of it. Source voltages of opposite signs are produced by motion in opposite directions, and the source voltages are also reversed by reversing poles. The same results are produced if the coil is moved rather than the magnet—it is the relative motion that is important. The faster the motion, the greater the source voltage, and there is no source voltage when the magnet is stationary relative to the coil.


7.3: Conceptualizing Structures of Power
A social structure is a set of long-lasting social relationships, practices and institutions that can be difficult to see at work in our daily lives. They are intangible social relations, but work much in the same way as structures we can see: buildings and skeletal systems are two examples. The human body is structured by bones; that is to say that the rest of our bodies’ organs and vessels are where they are because bones provide the structure upon which these other things can reside. Structures limit possibility, but they are not fundamentally unchangeable. For instance, our bones may deteriorate over time, suffer acute injuries, or be affected by disease, but they never spontaneously change location or disappear into thin air. Such is the way with social structures.
The elements of a social structure, the parts of social life that direct possible actions, are the institutions of society. These will be addressed in more detail later, but for now social institutions may be understood to include: the government, work, education, family, law, media, and medicine, among others. To say these institutions direct, or structure, possible social action, means that within the confines of these spaces, there are rules, norms, and procedures that limit what actions are possible. For instance, family is a concept near and dear to most, but historically and culturally, family forms have been highly defined and structured. According to Dorothy Smith (1993), the standard North American family includes two married, heterosexual parents and one or more biologically-related children. It also includes a division of labor, in which the husband/father earns a larger income and the wife/mother takes responsibility for most of the care-taking and childrearing. Although families vary in all sorts of ways, this is the norm to which they are most often compared. Thus, while we may consider our pets, friends, and lovers as family, the state, the legal system, and the media do not affirm these possibilities in the way they affirm the SNAF. In turn, when most people think of who is in their family, the normative notion of parents and children are the structures that they consider.
Overlaying these social structures are structures of power . By power we mean two things: 1) access to and through the various social institutions mentioned above, and 2) processes of privileging, normalizing, and valuing certain identities over others. This definition of power highlights the structural, institutional nature of power, while also highlighting the ways in which culture works in the creation and privileging of certain categories of people. Power in American society is organized along the axis of gender, race, class, sexuality, ability, age, nation, and religious identities. Some identities are more highly valued, or more normalized, than others—typically because they are contrasted to identities thought to be less valuable or less “normal.” Thus, identities are not only descriptors of individuals, but grant a certain amount of collective access to the institutions of social life. This is not to say, for instance, that all white people are alike and wield the same amount of power over all people of color. It does mean that white, middle-class women as a group tend to hold more social power than middle-class women of color. This is where the concept of intersectionality is key. All individuals have multiple aspects of identity, and simultaneously experience some privileges due to their socially valued identity statuses, and disadvantages due to their devalued identity statuses. Thus a white, heterosexual middle-class woman may be disadvantaged compared to a white middle-class man, but she may experience advantages in different contexts in relation to a black, heterosexual middle-class woman, or a white, heterosexual working-class man, or a white lesbian upper-class woman.
At the higher level of social structure, we can see that some people have greater access to resources and institutionalized power across the board than others. Sexism is the term we use for discrimination and blocked access women face. Genderism describes discrimination and blocked access that transgender and other non-binary people face. Racism describes discrimination and blocked access on the basis of race, which is based on socially-constructed meanings rather than biological differences. Classism describes discrimination on the basis of social class, or blocked access to material wealth and social status. Ableism describes discrimination on the basis of physical, mental, or emotional impairment or blocked access to the fulfillment of needs and in particular, full participation in social life. These “-isms” reflect dominant cultural notions that women, trans people, people of color, poor people, and disabled people are inferior to men, non-trans people, white people, middle- and upper-class people, and non-disabled people. Yet, the “-isms” are greater than individuals’ prejudice against women, trans people, people of color, the poor, and disabled people. For instance, in the founding of the United States the institutions of social life, including work, law, education, and the like, were built to benefit wealthy, white men since at the time these were, by law, the only real “citizens' ' of the country. Although these institutions have significantly changed over time in response to social movements and more progressive cultural shifts, their sexist, genderist, racist, classist, and ableist structures continue to persist in different forms today. Similar-sounding to “-isms,” the language of “-ization,” such as in “racialization,” is used to highlight the formation or processes by which these forms of difference have been given meaning and power (Omi and Winant 1986).
Just like the human body’s skeletal structure, social structures are not immutable, or completely resistant to change. Social movements mobilized on the basis of identities have fought for increased equality and changed the structures of society, in the US and abroad, over time. However, these struggles do not change society overnight; some struggles last decades, centuries, or remain infinitely unfinished. The structures and institutions of social life change slowly, but they can and do change based on the concerted efforts of individuals, social movements, and social institutions.


3: Getting Started with R
Robots are nice to work with.
–Roger Zelazny 13
In this chapter I’ll discuss how to get started in R. I’ll briefly talk about how to download and install R, but most of the chapter will be focused on getting you started typing R commands. Our goal in this chapter is not to learn any statistical concepts: we’re just trying to learn the basics of how R works and get comfortable interacting with the system. To do this, we’ll spend a bit of time using R as a simple calculator, since that’s the easiest thing to do with R. In doing so, you’ll get a bit of a feel for what it’s like to work in R. From there I’ll introduce some very basic programming ideas: in particular, I’ll talk about the idea of defining variables to store information, and a few things that you can do with these variables.
However, before going into any of the specifics, it’s worth talking a little about why you might want to use R at all. Given that you’re reading this, you’ve probably got your own reasons. However, if those reasons are “because that’s what my stats class uses”, it might be worth explaining a little why your lecturer has chosen to use R for the class. Of course, I don’t really know why other people choose R, so I’m really talking about why I use it.
- It’s sort of obvious, but worth saying anyway: doing your statistics on a computer is faster, easier and more powerful than doing statistics by hand. Computers excel at mindless repetitive tasks, and a lot of statistical calculations are both mindless and repetitive. For most people, the only reason to ever do statistical calculations with pencil and paper is for learning purposes. In my class I do occasionally suggest doing some calculations that way, but the only real value to it is pedagogical. It does help you to get a “feel” for statistics to do some calculations yourself, so it’s worth doing it once. But only once!
- Doing statistics in a spreadsheet (e.g., Microsoft Excel) is generally a bad idea in the long run. Although many people are likely feel more familiar with them, spreadsheets are very limited in terms of what analyses they allow you do. If you get into the habit of trying to do your real life data analysis using spreadsheets, then you’ve dug yourself into a very deep hole.
- Avoiding proprietary software is a very good idea. There are a lot of commercial packages out there that you can buy, some of which I like and some of which I don’t. They’re usually very glossy in their appearance, and generally very powerful (much more powerful than spreadsheets). However, they’re also very expensive: usually, the company sells “student versions” (crippled versions of the real thing) very cheaply; they sell full powered “educational versions” at a price that makes me wince; and they sell commercial licences with a staggeringly high price tag.  It’s hard to blame them for trying, but personally I’m not in favour of shelling out thousands of dollars if I can avoid it. And you can avoid it: if you make use of packages like R that are open source and free, you never get trapped having to pay exorbitant licensing fees.
- Something that you might not appreciate now, but will love later on if you do anything involving data analysis, is the fact that R is highly extensible. When you download and install R, you get all the basic “packages”, and those are very powerful on their own. However, because R is so open and so widely used, it’s become something of a standard tool in statistics, and so lots of people write their own packages that extend the system. And these are freely available too. One of the consequences of this, I’ve noticed, is that if you open up an advanced textbook (a recent one, that is) rather than introductory textbooks, is that a lot of them use R. In other words, if you learn how to do your basic statistics in R, then you’re a lot closer to being able to use the state of the art methods than you would be if you’d started out with a “simpler” system: so if you want to become a genuine expert in psychological data analysis, learning R is a very good use of your time.
- Related to the previous point: R is a real programming language. As you get better at using R for data analysis, you’re also learning to program. To some people this might seem like a bad thing, but in truth, programming is a core research skill across a lot of the social and behavioural sciences. Think about how many surveys and experiments are done online, or presented on computers. Think about all those online social environments which you might be interested in studying; and maybe collecting data from in an automated fashion. Think about artificial intelligence systems, computer vision and speech recognition. If any of these are things that you think you might want to be involved in – as someone “doing research in psychology”, that is – you’ll need to know a bit of programming. And if you don’t already know how to program, then learning how to do statistics using R is a nice way to start.
Those are the main reasons I use R. It’s not without its flaws: it’s not easy to learn, and it has a few very annoying quirks to it that we’re all pretty much stuck with, but on the whole I think the strengths outweigh the weakness; more so than any other option I’ve encountered so far.


3.5: Subspaces of Rp
\(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\twovec}[2]{\begin{pmatrix} #1 \\ #2 \end{pmatrix} } \) \(\newcommand{\threevec}[3]{\begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix} } \) \(\newcommand{\fourvec}[4]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \end{pmatrix} } \) \(\newcommand{\fivevec}[5]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \\ #5 \end{pmatrix} } \)
We saw that vectors in a basis for \(\mathbb R^p\) form the columns of an invertible matrix, which is necessarily a square matrix.
A basis for \(\mathbb R^p\) can be useful for it creates a coordinate system that helps us effectively navigate in \(\mathbb R^p\text{.}\) Sometimes, however, we find ourselves dealing with only a subset of \(\mathbb R^p\text{.}\) In particular, if we are given an \(m\times n\) matrix \(A\text{,}\) we have been interested in both the span of the columns of \(A\) and the solution space to the homogeneous equation \(A\mathbf x = \mathbf 0\text{.}\) In this section, we will expand the concept of basis to describe sets like these.
Preview Activity 3.5.1.
Let's consider the following matrix \(A\) and its reduced row echelon form.
- Are the columns of \(A\) linearly independent? Do they span \(\mathbb R^3\text{?}\)
- Give a parametric description of the solution space to the homogeneous equation \(A\mathbf x = \mathbf 0\text{.}\)
- Explain how this parametric description produces two vectors \(\mathbf w_1\) and \(\mathbf w_2\) whose span is the solution space to the equation \(A\mathbf x = \mathbf 0\text{.}\)
- What can you say about the linear independence of the set of vectors \(\mathbf w_1\) and \(\mathbf w_2\text{?}\)
- Let's denote the columns of \(A\) as \(\mathbf v_1\text{,}\) \(\mathbf v_2\text{,}\) \(\mathbf v_3\text{,}\) and \(\mathbf v_4\text{.}\) Explain why \(\mathbf v_3\) and \(\mathbf v_4\) can be written as linear combinations of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\)
- Explain why \(\mathbf v_1\) and \(\mathbf v_2\) are linearly independent and \(Span\{{\mathbf v_1,\mathbf v_2}\} = Span\{{\mathbf v_1, \mathbf v_2, \mathbf v_3, \mathbf v_4}\}\text{.}\)
Subspaces of \(\mathbb R^p\)
In the preview activity, we considered a \(3\times4\) matrix \(A\) and described two familiar sets of vectors. FIrst, we described the solution space to the homogeneous equation \(A\mathbf x = \mathbf 0\text{,}\) which is a set of vectors in \(\mathbb R^4\text{.}\) Next, we described the span of the columns of \(A\text{,}\) which is a set of vectors in \(\mathbb R^3\text{.}\) As we will see shortly, each of these sets has a common feature that we would like to study further: if we choose some vectors in one of these sets, any linear combination of those vectors is also in the set. This observation motivates the following definition.
A subspace of \(\mathbb R^p\) is a nonempty subset of \(\mathbb R^p\) such that any linear combination of vectors in that set is also in the set.
Without mentioning it explicitly, we have frequently encountered and worked with subspaces earlier in our investigations. Let's look at some examples to get comfortable with this concept.
It will be helpful to first look at some examples of subsets of \(\mathbb R^2\) that are not subspaces. First, consider the set of vectors in the first quadrant of \(\mathbb R^2\text{;}\) that is, vectors of the form \(\twovec{x}{y}\) where both \(x,y \geq 0\text{.}\) This subset is illustrated on the left of Figure 3.5.3.
If this subset were a subspace of \(\mathbb R^2\text{,}\) any linear combination of vectors in the first quadrant must also be in the first quadrant. If we consider the vector \(\mathbf v=\twovec{3}{2}\text{,}\) however, we can form the linear combination \(-\mathbf v=\twovec{-3}{-2}\text{,}\) which is not in the first quadrant, as seen on the right of Figure 3.5.3. Therefore, the set of vectors in the first quadrant is not a subspace.
This shows something important, however. Suppose that \(S\) is a subspace and \(\mathbf v\) is a vector in \(S\text{.}\) Any scalar multiple of \(\mathbf v\) is a linear combination of \(\mathbf v\) and so must be in \(S\) as well. This means that the line containing \(\mathbf v\) must be in \(S\text{.}\)
With this in mind, let's consider another example where we look at vectors that are in either the first or third quadrant; that is, we will consider vectors of the form \(\twovec{x}{y}\) where either \(x,y\geq 0\) or \(x,y\leq 0\text{,}\) as seen on the left of Figure 3.5.4.
If \(\mathbf v\) is a vector in this set, then the line containing \(\mathbf v\) is in the set. However, if we consider the vectors \(\mathbf v = \twovec{0}{3}\) and \(\mathbf w=\twovec{-2}{0}\text{,}\) then their sum \(\mathbf v+\mathbf w = \twovec{-2}{3}\) is not in the subset, as seen on the right of Figure 3.5.4. This subset is also not a subspace.
Let's look in \(\mathbb R^2\) and consider \(S\text{,}\) the set of vectors lying on the \(x\) axis; that is, vectors having the form \(\twovec{x}{0}\text{,}\) as shown on the left of Figure 3.5.6. Any scalar multiple of a vector lying on the \(x\) axis also lies on the \(x\) axis. Also, any sum of vectors lying on the \(x\) axis also lies on the \(x\) axis. Therefore, \(S\) is a subspace of \(\mathbb R^2\text{.}\) Notice that \(S\) is the span of the vector \(\twovec{1}{0}\text{.}\)
In fact, any line through the origin forms a subspace, as seen on the right of Figure 3.5.6. Indeed, any such line is the span of a nonzero vector on the line.
Activity 3.5.2.
We will look at some more subspaces of \(\mathbb R^2\text{.}\)
-
Explain why a line that does not pass through the origin, as seen to the right, is not a subspace of \(\mathbb R^2\text{.}\)
- Explain why any subspace of \(\mathbb R^2\) must contain the zero vector \(\mathbf 0\text{.}\)
- Explain why the subset \(S\) of \(\mathbb R^2\) that consists of only the zero vector \(\mathbf 0\) is a subspace of \(\mathbb R^2\text{.}\)
- Explain why the subspace \(S=\mathbb R^2\) is itself a subspace of \(\mathbb R^2\text{.}\)
- If \(\mathbf v\) and \(\mathbf w\) are two vectors in a subspace \(S\text{,}\) explain why \(Span\{{\mathbf v,\mathbf w}\}\) is contained in the subspace \(S\) as well.
-
Suppose that \(S\) is a subspace of \(\mathbb R^2\) containing two vectors \(\mathbf v\) and \(\mathbf w\) that are not scalar multiples of one another. What is the subspace \(S\) in this case?
This activity introduces an important idea. Suppose that we have a subspace \(S\) of \(\mathbb R^p\) and that vectors \(\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n\) are in \(S\text{.}\) We know that any linear combination of these vectors must also be in the subspace \(S\text{.}\) Since the span of these vectors is the set of all linear combinations of the vectors, it must be the case that \(Span\{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\}\) is in the subspace \(S\) as well.
With this in mind, we can list all the subspaces of \(\mathbb R^2\text{.}\) If a subspace \(S\) contains a nonzero vector, then it must contain the line containing that vector. If \(S\) contains two vectors \(\mathbf v\) and \(\mathbf w\) that are not scalar multiples of one another, then \(Span\{{\mathbf v,\mathbf w}\} = \mathbb R^2\) so the subspace \(S\) must be all of \(\mathbb R^2\text{.}\) These are the only possibilities:
- The subspace \(S=\{\mathbf 0\}\) consisting of only the zero vector .
- A line through the origin.
- The subspace \(S=\mathbb R^2\text{.}\)
Subspaces are the simplest subsets of \(\mathbb R^p\text{;}\) they are subsets in which we can perform the usual operations of scalar multiplication and vector addition without leaving the subset. Just as we can create bases for \(\mathbb R^p\text{,}\) we can create bases for subspaces as well.
A basis for a subspace \(S\) of \(\mathbb R^p\) is a set of vectors in \(S\) that are linearly independent and span \(S\text{.}\) It can be seen that any two bases have the same number of vectors. Therefore, we say that the dimension of the subspace \(S\text{,}\) denoted \(\dim S\text{,}\) is the number of vectors in any basis.
With this in mind, we can describe the possible spaces of \(\mathbb R^3\text{.}\)
- The subspace \(S=\{\mathbf 0\}\) is a subspace whose dimension is 0.
-
A line through the origin is a subspace whose dimension is 1. Any nonzero vector on the line forms a basis.
-
A plane through the origin is a subspace whose dimension is 2. For instance, the vectors \(\mathbf v_1\) and \(\mathbf v_2\) form a basis for the subspace shown here.
- Finally, the subspace \(S=\mathbb R^3\) is a subspace of \(\mathbb R^3\) whose dimension is 3.
Of course, there cannot be a subspace of \(\mathbb R^3\) whose dimension is four or higher since any set of four vectors in \(\mathbb R^3\) cannot be linearly independent.
We are most interested in two subspaces that are naturally associated with a matrix. With this background, we are now ready to introduce them.
The null space of \(A\)
When we looked at the linear independence of the columns of a matrix \(A\) in Section 2.4 , we were led to consider the homogeneous equation \(A\mathbf x = \mathbf 0\text{.}\) We note that this solution space forms a subspace that we call the null space of \(A\text{.}\)
If \(A\) is an \(m\times n\) matrix, we call the subset of vectors \(\mathbf x\) in \(\mathbb R^n\) satisfying \(A\mathbf x = \mathbf 0\) the null space of \(A\text{.}\) We denote it as \(Nul(A)\text{.}\)
The linearity of matrix multiplication, expressed in Proposition 2.2.3, tells us that \(Nul(A)\) is a subspace of \(\mathbb R^n\text{.}\) If \(\mathbf x_1\) and \(\mathbf x_2\) are both vectors in \(Nul(A)\text{,}\) we know that \(A\mathbf x_1 = \mathbf 0\) and \(A\mathbf x_2 = \mathbf 0\text{.}\) A linear combination of \(\mathbf x_1\) and \(\mathbf x_2\) can be written as \(c_1\mathbf x_1 + c_2\mathbf x_2\text{.}\) This linear combination is in \(Nul(A)\) because
Activity 3.5.3.
We will explore some null spaces in this activity.
-
Consider the matrix
\begin{equation*} A=\left[\begin{array}{rrr} 1 & 3 & -1 \\ -2 & 0 & -4 \\ 1 & 2 & 0 \\ \end{array}\right] \end{equation*}
and give a parametric description of the null space \(Nul(A)\text{.}\)
- Give a basis for and state the dimension of \(Nul(A)\text{.}\)
- The null space \(Nul(A)\) is a subspace of \(\mathbb R^p\) for which \(p\text{?}\)
-
Now consider the matrix \(A\) whose reduced row echelon form is given:
\begin{equation*} A \sim \left[\begin{array}{rrrr} 1 & 2 & 0 & -3 \\ 0 & 0 & 1 & 2 \\ \end{array}\right]\text{.} \end{equation*}
Give a parametric description of \(Nul(A)\text{.}\)
- Notice that the parametric description gives a set of vectors that span \(Nul(A)\text{.}\) Explain why this set of vectors is linearly independent and hence forms a basis. What is the dimension of \(Nul(A)\text{?}\)
- For this matrix, \(Nul(A)\) is a subspace of \(\mathbb R^p\) for what \(p\text{?}\)
- What is the relationship between the dimensions of the matrix \(A\text{,}\) the number of pivot positions of \(A\) and the dimension of \(Nul(A)\text{?}\)
- Suppose that the columns of a matrix \(A\) are linearly independent. What can you say about \(Nul(A)\text{?}\)
- If \(A\) is an invertible \(n\times n\) matrix, what can you say about \(Nul(A)\text{?}\)
- Suppose that \(A\) is a \(5\times 10\) matrix and that \(Nul(A) = \mathbb R^{10}\text{.}\) What can you say about the matrix \(A\text{?}\)
Let's consider an example of our own. Suppose we have a matrix \(A\) and its reduced row echelon form:
To find a parametric description of the solution space to \(A\mathbf x=\mathbf 0\text{,}\) imagine that we augment both \(A\) and its reduced row echelon form by a column of zeroes, which leads to the equations
Notice that \(x_3\text{,}\) \(x_4\text{,}\) and \(x_5\) are free variables so we rewrite these equations as
Writing this as a vector, we have
This expression says that any vector \(\mathbf x\) satisfying \(A\mathbf x= \mathbf 0\) is a linear combination of the vectors
It is easy to see that these vectors are linearly independent. Remember that we saw in Section 2.4 that this set of vectors is linearly dependent if any linear combination \(c_1\mathbf v_1 + c_2\mathbf v_2 + c_3\mathbf v_3 = \mathbf 0\) implies that \(c_1=c_2=c_3 = 0\text{.}\) But this linear combination would be
This expression shows that \(c_1=c_2=c_3=0\) so the vectors are linearly independent.
Therefore, we see that the vectors
form a basis for \(Nul(A)\) showing that \(Nul(A)\) is a three-dimensional subspace of \(\mathbb R^5\text{.}\)
Notice that the dimension of \(Nul(A)\) is equal to the number of free variables, which equals the number of columns of \(A\) minus the number of pivot positions. This example illustrates a general principle that motivates the following dimension.
The \(rank\) of a matrix \(A\text{,}\) denoted \(Rank(A)\text{,}\) is the number of pivot positions of \(A\text{.}\)
As illustrated by the previous example, if \(A\) is an \(m\times n\) matrix, then \(Nul(A)\) is a subspace of \(\mathbb R^n\) and
or
We may consider two extreme cases. If \(Nul(A)=\{\mathbf 0\}\text{,}\) then \(\dim~Nul(A) = 0\) so that \(Rank(A) = n\text{.}\) This means that the number of pivot positions is equal to the number of columns. In this case, there are no free variables in the description of the solutions to the homogeneous equation \(A\mathbf x = \mathbf 0\) so there is only the trivial solution. This is exactly what we are saying when we say that \(Nul(A) = \{\mathbf 0\}\text{.}\)
Similarly, if \(Nul(A) = \mathbb R^n\text{,}\) then \(\dim~Nul(A) = n\text{,}\) which implies that \(Rank(A) = 0\text{.}\) This means that \(A\) does not have any pivot positions and so \(A\) must be the zero matrix \(0\text{.}\) This is also consistent with what we already know: if \(Nul(A)=\mathbb R^n\text{,}\) then \(A\mathbf x = \mathbf 0\) for any vector \(\mathbf x\text{.}\) This can only be true if \(A = 0\text{.}\)
The column space of \(A\)
Besides the null space, the other subspace that is naturally associated to a matrix \(A\) is its column space.
If \(A\) is an \(m\times n\) matrix, we call the span of its columns the column space of \(A\) and denote it as \(Col(A)\text{.}\)
Notice that the columns of \(A\) are vectors in \(\mathbb R^m\text{,}\) which means that any linear combination of the columns is also in \(\mathbb R^m\text{.}\) The column space is therefore a subset of \(\mathbb R^m\text{.}\)
We can also see \(Col(A)\) is a subspace of \(\mathbb R^m\text{.}\) First, notice that a vector is in \(Col(A)\) if it is a linear combination of the columns of \(A\text{.}\) This means that \(\mathbf b\) is in \(Col(A)\) if there is a vector \(\mathbf x\) such that \(A\mathbf x = \mathbf b\text{.}\) To see that \(Col(A)\) is a subspace of \(\mathbb R^m\text{,}\) we need to check that any linear combination of vectors in \(Col(A)\) is also in \(Col(A)\text{.}\) This follows, once again, from the linearity of matrix multiplicaiton expressed in Proposition 2.2.3.
If vectors \(\mathbf b_1\) and \(\mathbf b_2\) are in \(Col(A)\text{,}\) then there are vectors \(\mathbf x_1\) and \(\mathbf x_2\) such that \(A\mathbf x_1 = \mathbf b_1\) and \(A\mathbf x_2 = \mathbf b_2\text{.}\) Therefore, if we have a linear combination of \(\mathbf b_1\) and \(\mathbf b_2\text{,}\) then
which shows that the linear combination is itself in the column space of \(A\text{.}\) Therefore, \(Col(A)\) is a subspace of \(\mathbb R^m\text{.}\)
Activity 3.5.4.
We will explore some column spaces in this activity.
-
Consider the matrix
\begin{equation*} A= \left[\begin{array}{rrr} \mathbf v_1 & \mathbf v_2 & \mathbf v_3 \end{array}\right] = \left[\begin{array}{rrr} 1 & 3 & -1 \\ -2 & 0 & -4 \\ 1 & 2 & 0 \\ \end{array}\right]\text{.} \end{equation*}
Since \(Col(A)\) is the span of the columns, the vectors \(\mathbf v_1\text{,}\) \(\mathbf v_2\text{,}\) and \(\mathbf v_3\) naturally span \(Col(A)\text{.}\) Are these vectors linearly independent?
- Show that \(\mathbf v_3\) can be written as a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\) Then explain why \(Col(A)=Span\{{\mathbf v_1,\mathbf v_2}\}\text{.}\)
- Explain why the vectors \(\mathbf v_1\) and \(\mathbf v_2\) form a basis for \(Col(A)\text{.}\) This shows that \(Col(A)\) is a 2-dimensional subspace of \(\mathbb R^2\) and is therefore a plane.
-
Now consider the matrix \(A\) and its reduced row echelon form:
\begin{equation*} A = \left[\begin{array}{rrrr} -2 & -4 & 0 & 6 \\ 1 & 2 & 0 & -3 \\ \end{array}\right] \sim \left[\begin{array}{rrrr} 1 & 2 & 0 & -3 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\text{.} \end{equation*}
We will call the columns \(\mathbf v_1\text{,}\) \(\mathbf v_2\text{,}\) \(\mathbf v_3\text{,}\) and \(\mathbf v_4\text{.}\) Explain why \(\mathbf v_2\text{,}\) \(\mathbf v_3\text{,}\) and \(\mathbf v_4\) can be written as a linear combination of \(\mathbf v_1\text{.}\)
- Explain why \(Col(A)\) is a 1-dimensional subspace of \(\mathbb R^2\) and is therefore a line.
- What is the relationship between the dimension \(\dim~Col(A)\) and the rank \(Rank(A)\text{?}\)
- What is the relationship between the dimension of the column space \(Col(A)\) and the null space \(Nul(A)\text{?}\)
- If \(A\) is an invertible \(9\times9\) matrix, what can you say about the column space \(Col(A)\text{?}\)
- If \(Col(A)=\{\mathbf 0\}\text{,}\) what can you say about the matrix \(A\text{?}\)
Once again, we will consider the matrix \(A\) and its reduced row echelon form:
We will denote the columns as \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_5\text{.}\)
It is certainly true that \(Col(A) = Span\{{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_5}\}\) by the definition of the column space. However, the reduced row echelon form of the matrix shows us that the vectors are not linearly independent so \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_5\) do not form a basis for \(Col(A)\text{.}\)
From the reduced row echelon form, however, we can see that
This means that any linear combination of \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_5\) can be written as a linear combination of just \(\mathbf v_1\) and \(\mathbf v_2\text{.}\) Therefore, we see that \(Col(A) = Span\{{\mathbf v_1,\mathbf v_2}\}\text{.}\) Moreover, the reduced row echelon form shows that \(\mathbf v_1\) and \(\mathbf v_2\) are linearly independent, which implies that they form a basis for \(Col(A)\text{.}\) Therefore, \(Col(A)\) is a 2-dimensional subspace of \(\mathbb R^3\text{,}\) which is a plane in \(\mathbb R^3\text{,}\) having basis
In general, a column without a pivot position can be written as a linear combination of the columns that have pivot positions. This means that a basis for \(Col(A)\) will always be given by the columns of \(A\) having pivot positions. Therefore, the dimension of the column space \(Col(A)\) equals the rank \(Rank(A)\text{:}\)
If \(A\) is an \(m\times n\) matrix, this also says that
If \(A\) has a pivot position in every row, then \(\dim~Col(A) = Rank(A) = m\text{.}\) This implies that \(Col(A)\) is an \(m\)-dimensional subspace of \(\mathbb R^m\) and therefore, \(Col(A) = \mathbb R^m\text{.}\) This agrees with our earlier explorations in which we found that the columns of a matrix span \(\mathbb R^m\) if there is a pivot in every row.
At the other extreme, suppose that \(\dim~Col(A) = 0\text{.}\) The matrix \(A\) then has no pivots, which means that \(A\) must be the zero matrix \(0\text{.}\)
Summary
Once again, we find ourselves revisiting our two fundamental questions, expressed in Question 1.4.2, concerning the existence and uniqueness of solutions to linear systems. The column space \(Col(A)\) contains all the vectors \(\mathbf b\) for which the equation \(A\mathbf x = \mathbf b\) is consistent. The null space \(Nul(A)\) describes the solution space to the equation \(A\mathbf x = \mathbf 0\text{,}\) and its dimension tells us whether this equation has a unique solution.
- A subset \(S\) of \(\mathbb R^p\) is a subspace of \(\mathbb R^p\) if any linear combination of vectors in \(S\) is also in \(S\text{.}\) This essentially means that we can perform the usual vector operations of scalar multiplication and vector addition without leaving \(S\text{.}\) A basis of a subspace \(S\) is a linearly indepedent set of vectors in \(S\) whose span is \(S\text{.}\)
- If \(A\) is an \(m\times n\) matrix, then its null space \(Nul(A)\) is the solution space to the homogeneous equation \(A\mathbf x = \mathbf 0\text{.}\) It is a subspace of \(\mathbb R^n\text{.}\)
- A basis for \(Nul(A)\) is found through a parametric description of the solution space of \(A\mathbf x = \mathbf 0\text{.}\) We see that \(\dim~Nul(A) = n - Rank(A)\text{.}\)
- The column space \(Col(A)\) is the span of the columns of \(A\) and forms a subspace of \(\mathbb R^m\text{.}\)
- A basis for \(Col(A)\) is found from the columns of \(A\) that have pivot positions. The dimension is therefore \(\dim~Col(A) = Rank(A)\text{.}\)
Exercises
Suppose that \(A\) and its reduced row echelon form are
- The null space \(Nul(A)\) is a subspace of \(\mathbb R^p\) for what \(p\text{?}\) The column space \(Col(A)\) is a subspace of \(\mathbb R^p\) for what \(p\text{?}\)
- What are the dimensions \(\dim~Nul(A)\) and \(\dim~Col(A)\text{?}\)
- Find a basis for the column space \(Col(A)\text{.}\)
- Find a basis for the null space \(Nul(A)\text{.}\)
Suppose that
- Is the vector \(\threevec{0}{-1}{-1}\) in \(Col(A)\text{?}\)
- Is the vector \(\fourvec{2}{1}{0}{2}\) in \(Col(A)\text{?}\)
- Is the vector \(\threevec{2}{-2}{0}\) in \(Nul(A)\text{?}\)
- Is the vector \(\fourvec{1}{-1}{3}{-1}\) in \(Nul(A)\text{?}\)
- Is the vector \(\fourvec{1}{0}{1}{-1}\) in \(Nul(A)\text{?}\)
Determine whether the following statements are true or false and provide a justification for your response. Unless otherwise stated, assume that \(A\) is an \(m\times n\) matrix.
- If \(A\) is a \(127\times 341\) matrix, then \(Nul(A)\) is a subspace of \(\mathbb R^{127}\text{.}\)
- If \(\dim~Nul(A) = 0\text{,}\) then the columns of \(A\) are linearly independent.
- If \(Col(A) = \mathbb R^m\text{,}\) then \(A\) is invertible.
- If \(A\) has a pivot position in every column, then \(Nul(A) = \mathbb R^m\text{.}\)
- If \(Col(A) = \mathbb R^m\) and \(Nul(A) = \{\mathbf 0\}\text{,}\) then \(A\) is invertible.
Explain why the following statements are true.
- If \(B\) is invertible, then \(Nul(BA) = Nul(A)\text{.}\)
- If \(B\) is invertible, then \(Col(AB) = Col(A)\text{.}\)
- If \(A\sim A'\text{,}\) then \(Nul(A) = Nul(A')\text{.}\)
For each of the following conditions, construct a \(3\times 3\) matrix having the given properties.
- \(\dim~Nul(A) = 0\text{.}\)
- \(\dim~Nul(A) = 1\text{.}\)
- \(\dim~Nul(A) = 2\text{.}\)
- \(\dim~Nul(A) = 3\text{.}\)
Suppose that \(A\) is a \(3\times 4\) matrix.
- Is it possible that \(\dim~Nul(A) = 0\text{?}\)
- If \(\dim~Nul(A) = 1\text{,}\) what can you say about \(Col(A)\text{?}\)
- If \(\dim~Nul(A) = 2\text{,}\) what can you say about \(Col(A)\text{?}\)
- If \(\dim~Nul(A) = 3\text{,}\) what can you say about \(Col(A)\text{?}\)
- If \(\dim~Nul(A) = 4\text{,}\) what can you say about \(Col(A)\text{?}\)
Consider the vectors
and suppose that \(A\) is a matrix such that \(Col(A)=Span\{{\mathbf v_1,\mathbf v_2}\}\) and \(Nul(A) = Span\{{\mathbf w_1,\mathbf w_2}\}\text{.}\)
- What are the dimensions of \(A\text{?}\)
- Find such a matrix \(A\text{.}\)
Suppose that \(A\) is an \(8\times 8\) matrix and that \(\det A = 14\text{.}\)
- What can you conclude about \(Nul(A)\text{?}\)
- What can you conclude about \(Col(A)\text{?}\)
Suppose that \(A\) is a matrix and there is an invertible matrix \(P\) such that
- What can you conclude about \(Nul(A)\text{?}\)
- What can you conclude about \(Col(A)\text{?}\)
In this section, we saw that the solution space to the homogeneous equation \(A\mathbf x = \mathbf 0\) is a subspace of \(\mathbb R^p\) for some \(p\text{.}\) In this exercise, we will investigate whether the solution space to another equation \(A\mathbf x = \mathbf b\) can form a subspace.
Let's consider the matrix
- Find a parametric description of the solution space to the homogeneous equation \(A\mathbf x = \mathbf 0\text{.}\)
-
Graph the solution space to the homogeneous equation to the right.
- Find a parametric description of the solution space to the equation \(A\mathbf x = \twovec{4}{-2}\) and graph it above.
- Is the solution space to the equation \(A\mathbf x = \twovec{4}{-2}\) a subspace of \(\mathbb R^2\text{?}\)
- Find a parametric description of the solution space to the equation \(A\mathbf x=\twovec{-8}{4}\) and graph it above.
- What can you say about all the solution spaces to equations of the form \(A\mathbf x = \mathbf b\) when \(\mathbf b\) is a vector in \(Col(A)\text{?}\)
- Suppose that the solution space to the equation \(A\mathbf x = \mathbf b\) forms a subspace. Explain why it must be true that \(\mathbf b = \mathbf 0\text{.}\)


9.25: Twin law
The twin law is the set of twin operations mapping two individuals of a twin. It is obtained by coset decomposition of the point group of the twin lattice with respect to the intersection group of the point groups of the individuals in their respective orientations. Each operation in the same coset is a possible twin operation that, from the lattice viewpoint, is equivalent to any other operation in the same coset. Any of these can be taken as coset representative and indicated by the symbol of the twin element: \(\bar 1\), [ uvw ] and ( hkl ) for the centre ( inversion twin ), direction of the rotation axis ( rotation twin ) and Miller indices of the mirror plane ( reflection twin ), in the order. Except when one refers to a specific plane or direction, the symbols { hkl } or < uvw > have to be be used to indicate all the planes or directions which belong to the same coset and are therefore equivalent under the point group of the individual.
In case of TLQS twinning the equivalence of the operations in a coset is only approximate.


2.2.E: Problems on Natural Numbers and Induction (Exercises)
Complete the missing details in Examples \((\mathrm{a}),(\mathrm{b}),\) and \((\mathrm{d})\).
Prove Theorem 2 in detail.
Suppose \(x_{k}<y_{k}, k=1,2, \ldots,\) in an ordered field. Prove by induction on \(n\) that
(a) \(\sum_{k=1}^{n} x_{k}<\sum_{k=1}^{n} y_{k}\)
(b) if all \(x_{k}, y_{k}\) are greater than zero, then
\[
\prod_{k=1}^{n} x_{k}<\prod_{k=1}^{n} y_{k}
\]
Prove by induction that
(i) \(1^{n}=1\);
(ii) \(a<b \Rightarrow a^{n}<b^{n}\) if \(a>0\).
Hence deduce that
(iii) \(0 \leq a^{n}<1\) if \(0 \leq a<1\);
(iv) \(a^{n}<b^{n} \Rightarrow a<b\) if \(b>0 ;\) proof by contradiction.
Prove the Bernoulli inequalities: For any element \(\varepsilon\) of an ordered field,
(i) \((1+\varepsilon)^{n} \geq 1+n \varepsilon\) if \(\varepsilon>-1\);
(ii) \((1-\varepsilon)^{n} \geq 1-n \varepsilon\) if \(\varepsilon<1 ; n=1,2,3, \ldots\)
For any field elements \(a, b\) and natural numbers \(m, n,\) prove that
\[
\begin{array}{ll}{\text { (i) } a^{m} a^{n}=a^{m+n} ;} & {\text { (ii) }\left(a^{m}\right)^{n}=a^{m n}} \\ {\text { (iii) }(a b)^{n}=a^{n} b^{n} ;} & {\text { (iv) }(m+n) a=m a+n a} \\ {\text { (v) } n(m a)=(n m) \cdot a ;} & {\text { (vi) } n(a+b)=n a+n b}\end{array}
\]
[Hint: For problems involving two natural numbers, fix \(m\) and use induction on \(n ]\).
Prove that in any field,
\[
a^{n+1}-b^{n+1}=(a-b) \sum_{k=0}^{n} a^{k} b^{n-k}, \quad n=1,2,3, \ldots
\]
Hence for \(r \neq 1\)
\[
\sum_{k=0}^{n} a r^{k}=a \frac{1-r^{n+1}}{1-r}
\]
(sum of \(n\) terms of a geometric series).
For \(n>0\) define
\[
\left(\begin{array}{l}{n} \\ {k}\end{array}\right)=\left\{\begin{array}{ll}{\frac{n !}{k !(n-k) !},} & {0 \leq k \leq n} \\ {0,} & {\text { otherwise }}\end{array}\right.
\]
Verify Pascal's law,
\[
\left(\begin{array}{l}{n+1} \\ {k+1}\end{array}\right)=\left(\begin{array}{l}{n} \\ {k}\end{array}\right)+\left(\begin{array}{c}{n} \\ {k+1}\end{array}\right).
\]
Then prove by induction on \(n\) that
(i) \((\forall k | 0 \leq k \leq n)\left(\begin{array}{l}{n} \\ {k}\end{array}\right) \in N ;\) and
(ii) for any field elements \(a\) and \(b\),
\[
(a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l}{n} \\ {k}\end{array}\right) a^{k} b^{n-k}, \quad n \in N \text { (the binomial theorem). }
\]
What value must \(0^{0}\) take for (ii) to hold for all \(a\) and \(b ?\)
Show by induction that in an ordered field \(F\) any finite sequence \(x_{1}, \ldots, x_{n}\) has a largest and a least term (which need not be \(x_{1}\) or \(x_{n} ) .\) Deduce that all of \(N\) is an infinite set, in any ordered field.
Prove in \(E^{1}\) that
(i) \(\sum_{k=1}^{n} k=\frac{1}{2} n(n+1)\);
(ii) \(\sum_{k=1}^{n} k^{2}=\frac{1}{6} n(n+1)(2 n+1)\);
(iii) \(\sum_{k=1}^{n} k^{3}=\frac{1}{4} n^{2}(n+1)^{2}\);
(iv) \(\sum_{k=1}^{n} k^{4}=\frac{1}{30} n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)\).


1.6: Account Wrap-Up
The discussion to this point has included all five types of accounts. Asset, Liability, and Stockholders’ Equity are accounts that appear on the balance sheet. Revenue and Expense are accounts reported on the income statement.
There is a hybrid version of these account types called contra accounts. The normal balance of a contra account is intentionally the opposite of the normal balance for a particular account classification. For example, a contra asset account has a credit balance instead of the normal debit balance for an asset account. A contra revenue account has a debit balance instead of the normal credit balance for a revenue account. This allows a company to continue to report an original amount by not making any changes directly to an account. Instead, an alternative contra account is used to report any changes. The original account and its contra account(s) are presented together on the financial statements to show original amount, total amount of changes, and the net result of the two (which is called carrying or net amount).
To recap, here is a list of the seven steps in the accounting cycle that we have covered to this point. Assume here that financial statements will be prepared at the end of each month.
| ACTION | WHEN | YOUR JOB | |
| 1. | Journalize transactions | Daily | THINK; analyze transactions |
| 2. | Post to ledgers | Daily | COPY from journal; CALCULATE |
| 3. | Income statement | End of month | COPY from ledgers; CALCULATE |
| 4. | Retained earnings statement | End of month | COPY from ledgers and income statement; CALCULATE |
| 5. | Balance sheet | End of month | COPY from ledgers and retained earnings statement; ADD |
| 6. | Journalize closing entries | End of month | THINK; same three entries |
| 7. | Post closing entries to ledgers | End of month | COPY from journal; CALCULATE |
The accounting cycle involves numerous steps, yet many of them are simple copying and calculating—procedures that may be tedious, but not difficult. For the seven steps in the accounting cycle discussed so far, five of them primarily involve only copying and/or calculating.
THINKING is involved when making journal entries—you have to analyze what is happening and translate the transaction into accounting language by selecting accounts to debit and credit. You often have to calculate amounts as well. This is involved in steps #1 and #6. However, closing entries are the same three every time, so they should become relatively routine.
The following table summarizes the rules of debit and credit and other facts about the accounts that you know so far:
| ACCOUNT TYPE | ACCOUNTS | TO INCREASE | TO DECREASE | NORMAL BALANCE | FINANCIAL STATEMENT | CLOSE OUT? |
| Asset |
Cash
Accounts Receivable Land Truck Equipment Building Furnishings |
debit | credit | debit | Balance Sheet | NO |
| Liability |
Accounts Payable
Note Payable |
credit | debit | credit | Balance Sheet | NO |
| Stockholders’ Equity |
Common Stock
|
credit | debit | credit | Balance Sheet | NO |
| Contra Stockholders’ Equity | Cash Dividends | credit | debit | credit |
Retained
Earnings Statement |
NO |
| Revenue | Fees Earned | credit | debit | credit | Income Statement | YES |
| Expense |
Wages Expense
Rent Expense Utilities Expense Supplies Expense Insurance Expense Advertising Expense Maintenance Expense Miscellaneous Expense |
debit | credit | debit | Income Statement | YES |


6.1: Lab 6- Respiratory System
Measurable Outcomes
- Understand and identify the anatomical structures of the respiratory system on available models.
- Deduce the pathway of air through the respiratory system.
- Determine the pathway of pulmonary circulation.
- Identify the various muscles involved in respiration.
- Recognize the hallmarks of lung histology.
- Demonstrate an adequate understand of the material in this section.
Background
The respiratory system is responsible for the gas exchange of oxygen and carbon dioxide. The main specialized organs of this process are the lungs which house clusters of sac-like structures known as alveoli . There are from 480 to 790 million alveoli which increase the efficiency of gas exchange by increasing surface area to around 118m 2 in men and 91m 2 in women. The respiratory system consists of the nasal cavity, pharynx, larynx, trachea, lungs, bronchi, bronchioles , and alveoli , along with their accessory structures. These structures are divided into the upper and lower respiratory systems , with the lower portion beginning at the larynx. The primary function of this system is to exchange oxygen and carbon dioxide between the body and the environment. Functionally, the respiratory system can be divided into the conducting zone , terminating at the terminal bronchioles; then air flows into the respiratory zone, where the actual gas exchange occurs.
Though we view each system individually in this lab, it is important to keep in mind that all organ systems overlap and work together in such a way that scientist are constantly discovering new connections. One such example is the nose . Not only is it the primary entrance and exit for respiration, but it also contains the olfactory epithelium, the primary structure of one of the special senses, olfaction. Likewise, the pharynx is a structure shared by both the respiratory and digestion systems.
Although both lungs functionally participate in respiration, they differ physically in various ways. The right lung is shorter and wider than the left lung, and the left lung occupies a smaller volume than the right. Another distinction between the two lungs is that the left lung contains the cardiac notch , which makes space for the heart. Furthermore, whereas the right lung has three lobes, the left lung has only two.
Though not visible on every model, each lung is surrounded by the pleura, which consists of two layers called the visceral and parietal pleurae. They are important because they lubricate the lungs and reduce friction during inhalation and exhalation.
Vocabulary for Respiratory System can be found on page(s) 169 .